and the function $\tau$ satisfies the kinematic equation(\ref{eq:POMeikonal}). Substituting approximation (\ref{eq:WKB}) intothe dynamic velocity continuation equation (\ref{eq:POMequation}),collecting the leading-order terms, and neglecting the $F$ functionleads to the partial differential equation for amplitude transport:\begin{equation}{\partial A \over \partial v} = v\,\tau\,\left(2\,{\partial A \over \partial x}\,{\partial \tau \over \partial x} + A\,{\partial^2 \tau \over \partial x^2}\right)\;.\label{eq:PAMPequation} \end{equation}The general solution of equation (\ref{eq:PAMPequation}) follows from thetheory of characteristics. It takes the form\begin{equation}A(x,v) = A(x_0,0)\,\exp{\left(\int_0^{v}\,u\,\tau(x,u)\,{\partial^2 \tau(x,u) \over \partial x^2}\,du\right)}\;,\label{eq:PAMPsolution} \end{equation}where the integral corresponds to thecurvilinear integration along the corresponding velocity ray, \new{and $x_0$corresponds to thestarting point of the ray.}In the case of a plane dipping reflector, the image of the reflector remainsplane in the velocity continuation process. Therefore, the secondtraveltime derivative ${\partial^2 \tau(x,u) \over \partial x^2}$ in(\ref{eq:PAMPsolution}) equals zero, and the exponential is equal toone. This means that the amplitude of the image does not change withthe velocity along the velocity rays. This fact does not agree with thetheory of conventional post-stack migration, which suggestsdownscaling the image by the ``cosine'' factor $\tau_0 \over\tau$ \cite[]{GEO46-05-07170733,Levin.sep.48.147}. The simplest way toinclude the cosine factor in the velocity continuation equation is toset the function $F$ to be ${1 \over t}\,{\partial P \over \partialv}$. The resulting differential equation\begin{equation}{{\partial^2 P} \over {\partial v\, \partial t}} +v\,t\,{{\partial^2 P} \over {\partial x^2}} +{1 \over t}\,{\partial P \over \partial v} = 0\label{eq:POMequation2} \end{equation}has the amplitude transport\begin{equation}A(x,v) = {\tau_0 \over \tau}\,A(x_0,0)\,\exp{\left(\int_0^{v}\,u\,\tau(x,u)\,{\partial^2 \tau(x,u) \over \partial x^2}\,du\right)}\;,\label{eq:PAMPsolution2} \end{equation}corresponding to the differential equation\begin{equation}{\partial A \over \partial v} = v\,\tau\,\left(2\,{\partial A \over \partial x}\,{\partial \tau \over \partial x} + A\,{\partial^2 \tau \over \partial x^2}\right) - A\,{1 \over \tau}\,{\partial \tau \over \partial v}\;.\label{eq:PAMPequation2} \end{equation}Appendix C proves that the time-and-space solution of the dynamicvelocity continuation equation (\ref{eq:POMequation2}) coincides with theconventional Kirchhoff migration operator.\begin{comment}The finite-difference implementation of zero-offset velocitycontinuation resembles the implementation of Claerbout's15-degree equation in a retarded coordinate system\cite[]{Claerbout.blackwell.76}. This implementation is discussed inmore detail in Appendix C. \end{comment}\subsection{Dynamics of Residual NMO}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%According to the theory of characteristics, described in the beginningof this section, the kinematic residual NMO equation(\ref{eq:ResNMOeikonal}) corresponds to the dynamic equation of the form\begin{equation}{{\partial P} \over {\partial v}} + {{h^2} \over {v^3\,t}}\,{{\partial P} \over {\partial t}} + F(h,t,v,P) = 0\label{eq:ResNMOdyn} \end{equation}\new{with the undetermined function $F$. In the case of $F=0$}, the general solutionis easily found to be\begin{equation}P(t,h,v) = \phi\left(t^2 + {h^2 \over v^2}\right)\;.\label{eq:ResNMOsol} \end{equation}where $\phi$ is an arbitrary smooth function.The combination of dynamic equations (\ref{eq:ResNMOdyn}) and(\ref{eq:POMequation2}) leads to an approximate prestack velocitycontinuation with the residual DMO effect neglected. To accomplish thecombination, one can simply add the term ${{h^2} \over{v^3\,t}}\,{{\partial^2 P} \over {\partial t^2}}$ fromequation~(\ref{eq:ResNMOdyn}) to the left-handside of equation (\ref{eq:POMequation2}). This addition changes thekinematics of velocity continuation, but does not change the amplitudeproperties embedded in the transport equation (\ref{eq:PAMPsolution2}).\cite{GEO38-04-06350642} and \cite{Hale.sepphd.36} \new{advocate using  an amplitude correction term in the NMO step. This term can be  easily added by selecting an appropriate function $F$ in  equation~(\ref{eq:ResNMOdyn}). The choice  $F=\frac{h^2}{v^3\,t^2}\,P$ results in the equation}\begin{equation}{{\partial P} \over {\partial v}} + {{h^2} \over {v^3\,t^2}}\,\left(t\,{{\partial P} \over {\partial t}} + P\right) = 0\label{eq:ResNMOdyn2} \end{equation}with the general solution\begin{equation}P(t,h,v) = \frac{1}{t}\,\phi\left(t^2 + {h^2 \over v^2}\right)\;,\label{eq:ResNMOsol} \end{equation}\new{which has the Dunkin-Levin amplitude correction term.}\subsection{Dynamics of Residual DMO}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%The case of residual DMO complicates the building of a dynamicequation because of the essential nonlinearity of the kinematicequation (\ref{eq:ResDMOeikonal}). One possible way to linearize theproblem is to increase the order of the equation. In this case, theresultant dynamic equation would include a term that has thesecond-order derivative with respect to velocity $v$. Such an equationdescribes two different modes of wave propagation and requiresadditional initial conditions to separate them. Another possible wayto linearize equation (\ref{eq:ResDMOeikonal}) is to approximate it atsmall dip angles.%For example, one%can obtain a recursively accurate approximation by a continued%fraction expansion of the square root in equation (\ref{eq:ResDMOeikonal}),%analogously to Muir's method in conventional finite-difference%migration \cite[]{Claerbout.blackwell.85}. In this case, the dynamicequation would contain only the first-order derivative with respect tothe velocity and high-order derivatives with respect to the otherparameters. The third, and probably the most attractive, method is tochange the domain of consideration. For example, one could switch fromthe common-offset domain to the domain of offset dip. Thismethod implies a transformation similar to slant stacking ofcommon-midpoint gathers in the post-migration domain in order toobtain the local offset dip information. Equation (\ref{eq:ResDMOeikonal})transforms, with the help of the results from Appendix A, to the form\begin{equation}v^3\,{{\partial \tau} \over {\partial v}} = {{\tau\,\sin^2{\theta}} \over{\cos^2{\alpha} - \sin^2{\theta}}}\;,\label{eq:noh} \end{equation}with\begin{equation}\cos^2{\alpha} = \left(1 + v^2 \,\left({{\partial \tau} \over {\partial x}}\right)^2\right)^{-1}\;,\label{eq:cos2a} \end{equation}and\begin{equation}\sin^2{\alpha} = v^2\,\left({{\partial \tau} \over {\partial h}}\right)^2\,\left(1 + v^2 \,\left({{\partial \tau} \over {\partial h}}\right)^2\right)^{-1}\;.\label{eq:sin2g} \end{equation}For a constant offset dip $\tan{\theta} = v\,{{\partial \tau} \over{\partial h}}$, the dynamic analog of equation (\ref{eq:noh}) is thethird-order partial differential equation\begin{equation}v\,  \cot^2{\theta}\,{{\partial^3 P} \over {\partial t^2\, \partial v}} -v^3\,{{\partial^3 P} \over {\partial x^2\, \partial v}}+ t\,{{\partial^3 P} \over {\partial t^2\, \partial v}} +v^2\,t\,{{\partial^3 P} \over {\partial x^2\, \partial t}} = 0\;.\label{eq:ResDMOdyn} \end{equation}Equation (\ref{eq:ResDMOdyn}) does not strictly comply with the theory ofsecond-order linear differential equations. Its properties andpractical applicability require further research.\section{Conclusions}%%%%%%%%%%%%%%%%%%%%%I have derived kinematic and dynamic equations for residual time migrationin the form of a continuous velocity continuation process. Thisderivation explicitly decomposes prestackvelocity continuation into three parts corresponding tozero-offset continuation, residual NMO, and residual DMO. These threeparts can be treated separately both for simplicity of theoreticalanalysis and for practical purposes. It is important to note that inthe case of a three-dimensional migration, all three components ofvelocity continuation have different dimensionality. Zero-offsetcontinuation is fully 3-D. It can be split into two 2-D continuationsin the in- and cross-line directions. Residual DMO is atwo-dimensional common-azimuth process. Residual NMO is a 1-Dsingle-trace procedure.The dynamic properties of zero-offset velocity continuation areprecisely equivalent to those of conventional post-stack migrationmethods such as Kirchhoff migration. Moreover, the Kirchhoff migrationoperator coincides with the integral solution of the velocitycontinuation differential equation for continuation from the zerovelocity plane.This rigorous theory of velocity continuation gives us new insights into themethods of prestack migration velocity analysis. Extensions to the case ofdepth migration in a variable velocity background are developed by\cite{hong} and \cite{adler}. \new{A practical application of  velocity continuation to migration velocity analysis is demonstrated in the  companion paper} \cite[]{second}, \new{where the general theory is used to  design efficient and practical algorithms.}\section{Acknowledgments}%%%%%%%%%%%%%%%%%%%%%%%%%This work was completed when the author was a member of the StanfordExploration Project (SEP) at Stanford University. The financialsupport was provided by the SEP sponsors.I thank Bee Bednar, Biondo Biondi, Jon Claerbout, Sergey Goldin, Bill Harlan,David Lumley, and Bill Symes for useful and stimulating discussions.\new{Paul Fowler, Hugh Geiger, Samuel Gray, and one anonymous reviewer  provided valuable suggestions that improved the quality of the paper.}\bibliographystyle{seg}\bibliography{SEP2,paper,spec,velcon,SEG}\append{DERIVING THE KINEMATIC EQUATIONS}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%The main goal of this appendix is to derive the partial differentialequation describing the image surface in adepth-midpoint-offset-velocity space.\inputdir{XFig}\sideplot{vlcray}{width=0.9\textwidth}{Reflection rays in a constant  velocity medium (a scheme).}The derivation starts with observing a simple geometry of reflectionin a constant-velocity medium, shown in Figure \ref{fig:vlcray}. Thewell-known equations for the apparent slowness\begin{equation}{{\partial t} \over {\partial s}} \,=\,{ {\sin{\alpha_1}} \over {v}}\;,\label{eq:snell1}\end{equation}\begin{equation}{{\partial t} \over {\partial r}} \,=\, {{\sin{\alpha_2}} \over {v}}  \label{eq:snell2}\end{equation} relate the first-order traveltime derivatives for the reflected wavesto the emergence angles of the incident and reflected rays. Here $s$stands for the source location at the surface, $r$ is the receiverlocation, $t$ is the reflection traveltime, $v$ is the constantvelocity, and $\alpha_1$ and $\alpha_2$ are the angles shown in Figure\ref{fig:vlcray}. Considering the traveltime derivative with respect tothe depth of the observation surface $z$ shows that thecontributions of the two branches of the reflected ray, addedtogether, form the equation\begin{equation}- {{\partial t} \over {\partial z}} \,=\,{{\cos{\alpha_1}} \over {v}} +{{\cos{\alpha_2}} \over {v}}\;.\label{eq:snell3}\end{equation}It is worth mentioning that the elimination of angles from equations(\ref{eq:snell1}), (\ref{eq:snell2}), and (\ref{eq:snell3}) leads tothe famous {\em double-square-root equation,}\begin{equation}- v\,{{\partial t} \over {\partial z}} \,=\,\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial s}}\right)^2} +\sqrt{1 -  v^2\,\left({{\partial t} \over {\partial r}}\right)^2}\;,\label{eq:DSR}\end{equation}published in the Russian literature by \cite{alekseev} and commonlyused in the form of a pseudo-differential dispersion relation\cite[]{Clayton.sep.14.21,Claerbout.blackwell.85} for prestackmigration \cite[]{Yilmaz.sepphd.18,Popovici.sep.84.53}. Consideredlocally, equation (\ref{eq:DSR}) is independent of the constant velocityassumption and enables \new{recursive} prestack downwardcontinuation of reflected waves in heterogeneous \new{isotropic}media.Introducing the midpoint coordinate $x = {{s+ r} \over 2}$ and half-offset$h = {{r - s} \over 2}$, one can apply the chain rule and elementarytrigonometric equalities to formulas (\ref{eq:snell1}) and(\ref{eq:snell2}) and transform these formulas to \begin{equation}{{\partial t} \over {\partial x}} \,=\, {{\partial t} \over {\partial s}} + {{\partial t} \over {\partial r}} \,=\, { {2 \sin{\alpha}\,\cos{\theta}} \over {v}}\;,\label{eq:snells1}\end{equation}\begin{equation}{{\partial t} \over {\partial h}} \,=\,{{\partial t} \over {\partial r}} - {{\partial t} \over {\partial s}} \,=\, { {2 \cos{\alpha}\,\sin{\theta}} \over {v}} \;,\label{eq:snells2}\end{equation}where $\alpha = {{\alpha_1 + \alpha_2} \over 2}$ is the dip angle, and$\theta = {{\alpha_2 - \alpha_1} \over 2}$ is the reflection angle\cite[]{Clayton.sep.14.21,Claerbout.blackwell.85}. Equation(\ref{eq:snell3}) transforms analogously to\begin{equation}- {{\partial t} \over {\partial z}} \,=\,{{2 \cos{\alpha} \cos{\theta}} \over {v}}\;. \label{eq:snells3}\end{equation}This form of equation (\ref{eq:snell3}) is used to describe the stretchingfactor of the waveform distortion in depth migration \cite[]{Tygel}.Dividing (\ref{eq:snells1}) and (\ref{eq:snells2}) by(\ref{eq:snells3}) leads to\begin{equation}{{\partial z} \over {\partial x}} \,=\,- \tan{\alpha}\;, \label{eq:snellz1}\end{equation}\begin{equation}{{\partial z} \over {\partial h}} \,=\,- \tan{\theta}\;.\label{eq:snellz2}\end{equation}Equation~(\ref{eq:snellz2}) is the basis of the angle-gather construction of\cite{sandf}.Substituting formulas (\ref{eq:snellz1}) and (\ref{eq:snellz2}) into equation(\ref{eq:snells3}) yields yet another form of the double-square-root equation:\begin{equation}- {{\partial t} \over {\partial z}} \,=\, {2 \over {v}}\,\left[\sqrt{1 + \left({\partial z} \over {\partial x}\right)^2}\,\sqrt{1 + \left({\partial z} \over {\partial h}\right)^2}\right]^{-1}\;, \label{eq:snellz3}\end{equation}which is analogous to the dispersion relationship of Stolt prestackmigration \cite[]{GEO43-01-00230048}. The law of sines in the triangle formed by the incident and reflectedray leads to the explicit relationship between the traveltime and theoffset:\begin{equation}v\,t = 2\,h\,  {{\cos{\alpha_1}+ \cos{\alpha_2}} \over\sin{\left(\alpha_2-\alpha_1\right)}} = 2\,h\,{\cos{\alpha} \over\sin{\theta}} \;.\label{eq:length} \end{equation}An algebraic combination of formulas (\ref{eq:length}), (\ref{eq:snells1}), and(\ref{eq:snells2}) forms the basic kinematic equation of the offsetcontinuation theory \cite[]{ofcon}:\begin{equation}{{\partial t} \over {\partial h}} \,\left(t^2 + {{4\,h^2} \over {v^2}}\right)\,=\,